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Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe. But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe.

My question is:

Where can I find a good reference for the last fact?

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In case it makes easier for you to find: The fact you are looking for, I believe, is credited to Mostowski (so you can search for: Mostowski's absoluteness). –  Burak May 10 at 17:18
    
@Burak: Many thanks. –  user38200 May 10 at 17:33

1 Answer 1

up vote 9 down vote accepted

This is an immediate consequence of the fact that every $\Pi^1_1$ statement is equivalent to the assertion that a certain relation is well-founded, and well-foundedness is absolute between transitive models. If a larger model thinks a relation is well-founded, then the smaller model must agree since it can have no infinite descending sequence. If a smaller model thinks a relation is well-founded, then it will have an ordinal ranking function, which will witness well-foundedness in the larger model.

This theorem is contained in every descriptive set theory book, such as the excellent one of Moschovakis's.

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